Generalized Andr\'asfai graphs and special Betti diagrams of edge ideals

TL;DR

This paper extends known results about Betti diagrams and regularity of edge ideals from complete and bipartite graphs to a broader family called Generalized Andre1sfai graphs, revealing similar algebraic properties.

Contribution

It generalizes the shape of Betti diagrams and regularity results from specific graphs to the entire family of Generalized Andre1sfai graphs.

Findings

Removing a Hamiltonian cycle yields edge ideals with regularity t+2.

Betti diagrams exhibit a generalized special shape.

Regularity and projective dimension are explicitly determined.

Abstract

Edge ideals of graphs were introduced by Villarreal in 1990, and have been the subject of many studies since then. In the same year, Fr\"oberg characterized edge ideals with regularity 2 in combinatorial terms. This result was generalized by Fern\'andez-Ramos and Gimenez to regularity 3 for bipartite graphs. A key ingredient in these results is the particular shape of the Betti diagrams of the edge ideals of the graphs obtained after removing a Hamiltonian cycle from either a complete graph $ K_k$ or a complete bipartite graph $K_{k,k}$. In this work, we consider the family of Generalized Andr\'asfai graphs ${\rm GA}(t,k)$ with $t\geq 1 $ and $k \geq 2$. This family extends the families of complete graphs, since $K_{k+1} = {\rm GA}(1,k)$, and complete bipartite $k$-regular graphs, since $K_{k,k} = {\rm GA}(2,k)$. We show that the results known for $ K_k$ and $ K_{k,k}$ can be naturally extended to this family. More precisely, when removing a suitable Hamiltonian cycle from ${\rm GA}(t,k)$, the resulting edge ideal has regularity $t+2$, projective dimension $t(k-2)$ and a Betti diagram exhibiting a generalized version of the same special shape.